THE ELEMENTS OE THE ORBIT OF A SATELLITE. 
891 
Thus the point B is a true minimum on the surface, whilst the point A is a 
maximum-minimum, being situated on a saddle-shaped part of the surface. 
The lines n—fL start from A and B, but one deviates from the ridge of maxima 
towards the ravine ; and the other branch deviates from the valley of minima by going 
up the slope on the side remote from the origin. 
This surface enables us to perfectly determine the stabilities of the circular orbit, 
when planet and satellite are moving as parts of a rigid body. 
The configuration B is obviously dynamically stable in all respects ; for any con¬ 
figuration represented by a point near B must degrade down to B. 
It is also clear that the configuration A is dynamically unstable, but the nature of 
the instability is complex. A displacement on the right-hand side of the ridge of 
maxima will cause the satellite to recede from the planet, because x must increase 
when the point slides down hill. 
If the viscosity be small, the ellipticity given to the orbit will diminish, because A 
is not comprised between the two chain-dot curves. Thus for this class of tide the 
circularity is stable, whilst the configuration is unstable. 
A displacement on the left-hand side of the ridge of maxima will cause the satellite 
to fall into the planet, because the point will slide down into the ravine. But the 
circularity of the orbit is again stable. 
This figure at once shows that if planet and satellite be revolving with maximum 
energy as parts of a rigid body, and if, without altering the total moment of 
momentum, or the ecjuality of the two periods, we impart infinitesimal ellipticity to 
the orbit, the satellite will fall into the planet. This follows from the fact that the 
line n=S 2 runs on to the slope of the ravine. 
If on the other hand without affecting the moment of momentum, or the circularity, 
we infinitesimally disturb the relation n=fi, then the satellite will either recede or 
approach the planet according to the nature of the disturbance. 
These two statements are independent of the nature of the frictional interaction of 
the two bodies. 
The only parts of this figure which postulate anything about the nature of the inter¬ 
action are the curves ft = yf/2. 
I have not thought it worth while to illustrate the case where h is less than 4/3% 
or the negative side of the surface of energy; but both illustrations may easily be 
carried out. 
